| L(s) = 1 | − 1.85·3-s − 3.75·5-s − 3.33·7-s + 0.442·9-s + 2.31·11-s − 1.30·13-s + 6.97·15-s − 17-s − 1.32·19-s + 6.18·21-s − 2.50·23-s + 9.11·25-s + 4.74·27-s + 6.82·29-s + 7.79·31-s − 4.29·33-s + 12.5·35-s + 2.02·37-s + 2.41·39-s + 6.45·41-s − 4.87·43-s − 1.66·45-s − 7.41·47-s + 4.11·49-s + 1.85·51-s − 0.619·53-s − 8.70·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s − 1.68·5-s − 1.25·7-s + 0.147·9-s + 0.698·11-s − 0.361·13-s + 1.80·15-s − 0.242·17-s − 0.303·19-s + 1.34·21-s − 0.522·23-s + 1.82·25-s + 0.913·27-s + 1.26·29-s + 1.39·31-s − 0.747·33-s + 2.11·35-s + 0.333·37-s + 0.386·39-s + 1.00·41-s − 0.743·43-s − 0.247·45-s − 1.08·47-s + 0.587·49-s + 0.259·51-s − 0.0851·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.85T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 - 2.31T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 - 2.02T + 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 0.619T + 53T^{2} \) |
| 61 | \( 1 + 1.42T + 61T^{2} \) |
| 67 | \( 1 - 0.724T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 1.49T + 83T^{2} \) |
| 89 | \( 1 + 6.21T + 89T^{2} \) |
| 97 | \( 1 + 4.07T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.098967204245706258452212136519, −7.17128783733169244854039355075, −6.48645510473880959957755068491, −6.17023473403800097771804323413, −4.90870586650856285750853312905, −4.34386943371512007402842031737, −3.51756599128142099077613142950, −2.74291983693506004095876174417, −0.844065227341930856662468257343, 0,
0.844065227341930856662468257343, 2.74291983693506004095876174417, 3.51756599128142099077613142950, 4.34386943371512007402842031737, 4.90870586650856285750853312905, 6.17023473403800097771804323413, 6.48645510473880959957755068491, 7.17128783733169244854039355075, 8.098967204245706258452212136519
